**3. Results**

Table 1 shows the initial conditions of the vector Q0 which contains the initial conditions of the system, and the first vector of fluxes calculated by multiplying Q0 by the source/destination matrix A, which is contained in Table 2. The characteristic equation of this matrix is: 0 = 1.0x10 – 0.186x9 + (9.34x10-3)x8 + (9.50x10-5)x7 + (3.78x10-7)x6 + (6.45x10-10 )x5 + (4.18x10-13)x4 + (7.95x10-17)x3 + (7.88x10-22)x2 + (2.15x10-28)x + 0.


herein were conducted using MATLAB.

inputs to the model because the animal is fasting.

(4.18x10-13)x4 + (7.95x10-17)x3 + (7.88x10-22)x2 + (2.15x10-28)x + 0.

**3. Results** 

determinant (Δ) of the matrix. The trace is the sum of the eigenvalues, and the determinant in the product of the eigenvalues (Heinrich et al. 1977). A dimensionless τ-Δ parameter plane may be envisioned which relates the magnitude of these two parameters to model stability, or the type of instability (Edelstein-Keshet 1988). All matrix calculations described

**Figure 1.** A 10-compartment flow diagram of metabolite flux in fasting northern elephant seals as simulated in this study. The boxes (or pools) represent the moles of carbon present in the animal as each metabolite, and the arrows represent the interchange of carbon between metabolites. There are no

Table 1 shows the initial conditions of the vector Q0 which contains the initial conditions of the system, and the first vector of fluxes calculated by multiplying Q0 by the source/destination matrix A, which is contained in Table 2. The characteristic equation of this matrix is: 0 = 1.0x10 – 0.186x9 + (9.34x10-3)x8 + (9.50x10-5)x7 + (3.78x10-7)x6 + (6.45x10-10 )x5 + **Table 1.** Initial condition values in moles of carbon for the 10 compartment model shown in Figure 1. The flux is the sum of the flows into and out of each pool during the first time step.


**Table 2.** Source/destination matrix used to simulate the kinetics of metabolite flux in fasting northern elephant seal pups. Non-diagonal elements represent the rate constants for flow from the column pool to the row pool, in terms of time-1. The diagonal elements are the sum all rate constants for flows from the pool, i.e. the sum of the column.

The roots of this equation are the eigenvalues (λ) of the matrix: 0.00; -8.10x10-4; -4.89x10-3 ; -2.70x10-3; -3.25x10-3; -1.06x10-1; -1.02x10-5; -3.00x10-4; -6.80x10-2; -2.81x10-7. Notice that there is one zero value, indicating that this is a closed system. The remainder of the eigenvalues are all negative, indicating a stable system. The trace (τ), or sum of the eigenvalues, of matrix A is -0.186 and the determinant (Δ), or product of the eigenvalues of matrix A is zero. Figure 2 shows the dimensionless τ-Δ parameter plane. The point for matrix A would lie on the yaxis, just below the x-y intercept, indicating that the system approaches a saddle point of stability. Notice that τ2 is 0.0346, which is greater than 4Δ, which is zero, indicating again that the matrix lies in the portion of the phase-plane corresponding to a saddle point condition (Heinrich et al. 1977).

A Matrix Model of Fasting Metabolism in Northern Elephant Seal Pups 39

stable, they both declined at the same rate, which was equivalent to the rate of decline of the glycerol pool. This similarity in rates of decline suggests that these pools are closely linked through recycling pathways, and may in fact represent a subsystem of the

**Figure 3.** Linear plot of the changes in the size of the adipose tissue and sink pools over an 8-week simulation. These pools are plotted together because they were of similar size. As expected, the

Figure 6 shows the changes in nitrogen containing pools over an 8-week simulation. The tissue protein pool declined at a very slow rate, suggesting almost no protein catabolism. This is supported by the decline and continued low level of the urea and alanine pools. The continued low urea level indicates again that protein catabolism is occurring at a very low

Figure 7 shows the time course of the contents of the palmitate, glycerol, and ketone pools over a 24-hour simulation. Figure 8 shows the time course of the contents of the alanine, urea, and tissue protein pools over an 8-week simulation. Notice that the tissue protein pool

adipose tissue pool declined at a constant rate, while the sink pool accumulated carbon

model.

asymptotically.

rate.

**Figure 2.** Dimensionless Cartesian phase-plane representing the range of trace (τ) and determinant (Δ) values possible, and the regions of qualitative behavior of linear systems. The point corresponding to the system simulated here lies on the y-axis just to the left of the x-y intercept.

Figure 3 shows the changes in the size of the adipose tissue and sink pools over an 8-week simulation. These pools are plotted together because they were of similar size. As expected, the adipose tissue pool declined at a constant rate, while the sink pool accumulated carbon asymptotically.

Figure 4 shows the changes in the size of the various lipid and lipid-derived pools over an 8 week simulation. The ketone pool rose slightly early on, and then declined slowly, as did the palmitate pool. At this resolution, it appeared that the glycerol pool declined at a constant rate throughout the simulation but when compared to the glucose and lactate simulations (Figure 5) it became clear that this pool was poorly initialized.

Figure 5 shows the changes in carbohydrate and carbohydrate-derived pools over an 8-week simulation. Apparently the glucose and lactate pools were initialized incorrectly, as reflected in their rapid increase in the first week of the simulation. Once they became stable, they both declined at the same rate, which was equivalent to the rate of decline of the glycerol pool. This similarity in rates of decline suggests that these pools are closely linked through recycling pathways, and may in fact represent a subsystem of the model.

38 New Approaches to the Study of Marine Mammals

**Figure 2.** Dimensionless Cartesian phase-plane representing the range of trace (τ) and determinant (Δ) values possible, and the regions of qualitative behavior of linear systems. The point corresponding to

Figure 3 shows the changes in the size of the adipose tissue and sink pools over an 8-week simulation. These pools are plotted together because they were of similar size. As expected, the adipose tissue pool declined at a constant rate, while the sink pool accumulated carbon

Figure 4 shows the changes in the size of the various lipid and lipid-derived pools over an 8 week simulation. The ketone pool rose slightly early on, and then declined slowly, as did the palmitate pool. At this resolution, it appeared that the glycerol pool declined at a constant rate throughout the simulation but when compared to the glucose and lactate simulations

Figure 5 shows the changes in carbohydrate and carbohydrate-derived pools over an 8-week simulation. Apparently the glucose and lactate pools were initialized incorrectly, as reflected in their rapid increase in the first week of the simulation. Once they became

the system simulated here lies on the y-axis just to the left of the x-y intercept.

(Figure 5) it became clear that this pool was poorly initialized.

asymptotically.

**Figure 3.** Linear plot of the changes in the size of the adipose tissue and sink pools over an 8-week simulation. These pools are plotted together because they were of similar size. As expected, the adipose tissue pool declined at a constant rate, while the sink pool accumulated carbon asymptotically.

Figure 6 shows the changes in nitrogen containing pools over an 8-week simulation. The tissue protein pool declined at a very slow rate, suggesting almost no protein catabolism. This is supported by the decline and continued low level of the urea and alanine pools. The continued low urea level indicates again that protein catabolism is occurring at a very low rate.

Figure 7 shows the time course of the contents of the palmitate, glycerol, and ketone pools over a 24-hour simulation. Figure 8 shows the time course of the contents of the alanine, urea, and tissue protein pools over an 8-week simulation. Notice that the tissue protein pool declined almost imperceptibly, while the urea and alanine pools declined initially and then became stable.

A Matrix Model of Fasting Metabolism in Northern Elephant Seal Pups 41

**Figure 5.** Linear plot of the changes in carbohydrate and carbohydrate-derived pool over an 8-week simulation. Apparently the glucose and lactate pools were initialized incorrectly, as reflected in their rapid increase in the first week of the simulation. Once they became stable, they both declined at the same rate, which was equivalent to the rate of decline of the glycerol pool. This similarity in rates of decline suggests that these pools are closely linked through recycling pathways, and may in fact

represent a subsystem of the model.

Figure 9 shows the time course of the contents of the glucose, glycerol, and lactate pools over a 24-hour simulation. The glucose and lactate pools were apparently poorly parameterized, and grew rapidly to a zenith, and then declined steadily. The glycerol pool declined continuously.

Figure 10 shows the time course of the contents of the glucose, glycerol, lactate, alanine and ketones pools after re-estimation of the initial conditions by extrapolating the linear parts of Figures 7, 8 and 9 to time zero.

**Figure 4.** Log-normal plot of the changes in the size of the various lipid and lipid-derived pools over an 8-week simulation. The ketone pool rose slightly early on, and then declined slowly, as did the palmitate pool. The glycerol pool appeared to decline at a constant rate throughout the simulation.

became stable.

simulation.

declined continuously.

Figures 7, 8 and 9 to time zero.

declined almost imperceptibly, while the urea and alanine pools declined initially and then

Figure 9 shows the time course of the contents of the glucose, glycerol, and lactate pools over a 24-hour simulation. The glucose and lactate pools were apparently poorly parameterized, and grew rapidly to a zenith, and then declined steadily. The glycerol pool

Figure 10 shows the time course of the contents of the glucose, glycerol, lactate, alanine and ketones pools after re-estimation of the initial conditions by extrapolating the linear parts of

**Figure 4.** Log-normal plot of the changes in the size of the various lipid and lipid-derived pools over an 8-week simulation. The ketone pool rose slightly early on, and then declined slowly, as did

the palmitate pool. The glycerol pool appeared to decline at a constant rate throughout the

**Figure 5.** Linear plot of the changes in carbohydrate and carbohydrate-derived pool over an 8-week simulation. Apparently the glucose and lactate pools were initialized incorrectly, as reflected in their rapid increase in the first week of the simulation. Once they became stable, they both declined at the same rate, which was equivalent to the rate of decline of the glycerol pool. This similarity in rates of decline suggests that these pools are closely linked through recycling pathways, and may in fact represent a subsystem of the model.

A Matrix Model of Fasting Metabolism in Northern Elephant Seal Pups 43

**Figure 7.** Linear plot of the changes in the size of the various lipid and lipid-derived pools over a 24 hour simulation. The ketone pool rose slightly early on, and then declined slowly, as did the palmitate

**Nitrogen Compunds Over A One Day Simulation**

**Urea Protein Alanine**

Hour 0 Hour 4 Hour 8 Hour 12 Hour 16 Hour 20 Hour 24 **Hours**

**Lipids and Glycerol In A One Day Simulation**

**Figure 8.** Changes in nitrogen containing pools over time. The tissue protein pool declined at a very slow rate, suggesting almost no protein catabolism. This is supported by the decline and continued low

Hour 0 Hour 4 Hour 8 Hour 12 Hour 16 Hour 20 Hour 24 **Hour**

level of the urea and alanine pools. The continued low urea level indicates again that protein

catabolism is occurring at a very low rate.

0.001

0.0001

0.001

0.01

0.1

**Carbon**

1

10

100

1000

0.01

0.1

1

**Palmitate**

**Ketones**

**Glycerol**

10

pool. The glycerol pool declined at a constant rate throughout the simulation.

**Figure 6.** Changes in nitrogen containing pools over an 8-week simulation. The tissue protein pool declined at a very slow rate, suggesting almost no protein catabolism. This is supported by the decline and continued low level of the urea and alanine pools. The continued low urea level indicates again that protein catabolism is occurring at a very low rate.

**Figure 6.** Changes in nitrogen containing pools over an 8-week simulation. The tissue protein pool declined at a very slow rate, suggesting almost no protein catabolism. This is supported by the decline and continued low level of the urea and alanine pools. The continued low urea level indicates again

that protein catabolism is occurring at a very low rate.

**Figure 7.** Linear plot of the changes in the size of the various lipid and lipid-derived pools over a 24 hour simulation. The ketone pool rose slightly early on, and then declined slowly, as did the palmitate pool. The glycerol pool declined at a constant rate throughout the simulation.

**Figure 8.** Changes in nitrogen containing pools over time. The tissue protein pool declined at a very slow rate, suggesting almost no protein catabolism. This is supported by the decline and continued low level of the urea and alanine pools. The continued low urea level indicates again that protein catabolism is occurring at a very low rate.

A Matrix Model of Fasting Metabolism in Northern Elephant Seal Pups 45

As initially formulated, the value of the rate constant for efflux from the sink pool (k10,10) was set at zero, because there should be no efflux from the sink. Eigenanalysis of this matrix yielded 9 negative eigenvalues and one zero eigenvalue, indicating a stable, closed system (Keith 1999). An eigenvalue of zero is to be expected from a matrix with a zero on the main diagonal (Edelstein-Keshet 1988, Swartzman 1987). For this reason, the zero eigenvalue can be considered an artifact of matrix construction, and not truly representative of the system being simulated. Therefore, a second analysis was conducted in which the value of the rate constant for efflux from the sink pool was set to one, which allowed all of the contents of the sink pool to exit at every time step. Eigenanalysis of this matrix yielded 10 negative eigenvalues, indicative of a now open and still stable system. The trace (τ) of this matrix was -0.186, and the determinant (Δ) was 2.174 x 10-32. In this case τ2 was 0.0346 and 4Δ was 8.696 x 10-32. Taken together, these values indicate that the system without a sink lies near a stable node condition in the phase-plane. However, this is difficult to reconcile with the biological reality that a fasting elephant seal with no food or water inputs is not at equilibrium, and

The genesis of this apparent contradiction may lie in differences in time scale or time constants. Differential equations with widely different time scales are "stiff" (Heinrich et al. 1977) and will have eigenvalues of different orders of magnitude. This is apparent here where the eigenvalues range from -1.06 x 10-1 to -2.81 x 10-7. The reciprocals of the eigenvalues are the relaxation times (Heinrich et al. 1977) and these likewise vary over six orders of magnitude, indicating that there are fast-reacting variables and slow-reacting variables in the simulation. Such hierarchical time structure may obscure predictions of model stability because the eigenvalues only characterize the system in the close time-neighborhood of the steady state where linear approximation is appropriate (Heinrich et al. 1977). Thus, predictions of model stability based on the signs of the eigenvalues may contradict a prediction of model instability based on relaxation times and slow-moving versus fast-moving variables in the system

Elevated palmitate levels are consistent with field data indicating that the major energy substrate during fasting (Castellini, *et. al.* 1987, Keith 1984). The decline in ketone levels through the simulation is consistent with field data indicating low levels of ketone bodies in the plasma of fasting northern elephant seal pups (Costa and Ortiz 1982). Declines in alanine, tissue protein, and urea levels are also consistent with field data (Pernia, *et. al.* 1980), and are validated by data which show that the lean body mass of the animal doesn't change during significantly during the fasting period (Ortiz, *et. al.* 1978). Lack of significant protein catabolism, with concomitant low urea levels, indicates that the animals do not maintain their elevated blood glucose levels at the expense of gluconeogenesis from amino acids (Keith 1984). Close similarities in the rates of decline of the glucose, glycerol, and lactate pools in the later parts of the simulation may be indicative of the extensive glucose carbon recycling which occurs in these animals during fasting. There is extensive interchange of carbon between these three pools, as indicated by high levels of Cori cycle and glucoseglycerol cycle activity (Keith 1984, Keith and Ortiz 1989). It is postulated that this high

(Heinrich et al. 1977) over the duration of the actual fast of the animal.

**4. Discussion** 

cannot survive forever (Ortiz *et. al.* 1978).

**Figure 9.** Changes in carbohydrate and carbohydrate-derived pool over 24 hours. Apparently the glucose and lactate pools were initialized incorrectly, as reflected in their rapid increase in the first hours of the simulation. Once they became stable, they both declined at the same rate, which was equivalent to the rate of decline of the glycerol pool. This similarity in rates of decline suggests that these pools are closely linked through recycling pathways, and may in fact represent a subsystem of the model.

**Figure 10.** Changes in the glucose, glycerol, lactate, alanine and ketones pools over 24 hours after reestimation of the initial conditions by extrapolating the linear parts of Figures 6, 7, and 8 to time zero.
